U.S. patent number 7,253,761 [Application Number 11/268,532] was granted by the patent office on 2007-08-07 for analog to digital conversion with signal expansion.
This patent grant is currently assigned to United States of America as Represented by the Secretary of the Army. Invention is credited to Gonzalo R. Arce, Sebastian Hoyos, Brian M Sadler.
United States Patent |
7,253,761 |
Hoyos , et al. |
August 7, 2007 |
Analog to digital conversion with signal expansion
Abstract
Included are embodiments of a method for converting a received
analog signal into a digital signal. Some embodiments of the method
can include receiving an analog signal; periodically dividing the
received analog signal into a plurality of discrete signals at a
predetermined interval, wherein each of the plurality of divided
signals is associated with a voltage; and quantizing the voltage
associated with at least one of the plurality of divided signals.
Other systems and methods are also provided.
Inventors: |
Hoyos; Sebastian (Berkeley,
CA), Sadler; Brian M (Laurel, MD), Arce; Gonzalo R.
(Newark, DE) |
Assignee: |
United States of America as
Represented by the Secretary of the Army (Washington,
DC)
|
Family
ID: |
38324365 |
Appl.
No.: |
11/268,532 |
Filed: |
November 8, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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60630762 |
Nov 8, 2004 |
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Current U.S.
Class: |
341/155;
341/143 |
Current CPC
Class: |
H03M
1/121 (20130101); H04B 1/71632 (20130101) |
Current International
Class: |
H03M
1/12 (20060101) |
Field of
Search: |
;341/155 |
References Cited
[Referenced By]
U.S. Patent Documents
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5568142 |
October 1996 |
Velazquez et al. |
6177893 |
January 2001 |
Velazquez et al. |
6252535 |
June 2001 |
Kober et al. |
6476749 |
November 2002 |
Yeap et al. |
6518908 |
February 2003 |
Boehm et al. |
|
Primary Examiner: Nguyen; Khai M.
Attorney, Agent or Firm: Adams; William V.
Parent Case Text
CROSS REFERENCE
This application claims the benefit of U.S. Provisional Application
No. 60/630,762 filed Nov. 8, 2004, which is hereby incorporated by
reference in its entirety.
Claims
The invention claimed is:
1. A method for converting a received analog signal into a digital
signal, the method comprising: receiving an analog signal;
periodically dividing the received analog signal into a plurality
of discrete signals at a predetermined interval, wherein each of
the plurality of divided signals is associated with a voltage; and
quantizing the voltage associated with at least one of the
plurality of divided signals using a vector quantizer with relaxed
characteristics operating over a digital conversion level that can
change with a speed lower than the Nyquist rate required for time
domain analog to digital conversion.
2. The method of claim 1, wherein dividing the received analog
signal includes a frequency domain basis expansion and wherein the
plurality of discrete signals are taken at a rate that guarantees
no aliasing in a discrete version of the signal.
3. The method of claim 1, wherein dividing the received analog
signal includes subjecting at least one of the plurality of divided
signals to at least one exponential.
4. The method of claim 1, further comprising introducing a
frequency offset.
5. The method of claim 1, wherein dividing the received analog
signal includes subjecting at least one of the plurality of divided
analog signals to an integral function.
6. The method of claim 1, further comprising filtering the received
analog signal.
7. The method of claim 1, wherein dividing the received analog
signal includes dividing using a plurality of orthogonal basis
functions.
8. A system for converting a received analog signal to a digital
signal, the system comprising: at least one receiving component
configured to receive an analog signal; at least one dividing
component configured to periodically divide the received analog
signal into a plurality of discrete signals at a predetermined
interval, wherein each of the plurality of divided signals is
associated with a voltage; and at least one quantizer configured to
quantize the voltage associated with at least one of the plurality
of divided signals using a vector quantizer with relaxed
characteristics operating over a digital conversion level that can
change with a speed lower than the Nyquist rate required for time
domain analog to digital conversion.
9. The analog to digital converter of claim 8, wherein the means
for dividing the received analog signal includes means for
providing a frequency domain basis and wherein the plurality of
discrete signals are taken at a rate that guarantees no aliasing in
a discrete version of the signal.
10. The analog to digital converter of claim 8, wherein the
dividing component includes means for subjecting at least one of
the plurality of divided signals to at least one exponential.
11. The analog to digital converter of claim 8, further comprising
means for introducing a frequency offset.
12. The analog to digital converter of claim 8, wherein dividing
component includes means for subjecting at least one of the
plurality of divided analog signals to an integral function.
13. The analog to digital converter of claim 8, further comprising
at least one filter for filtering the received analog signal.
14. The analog to digital converter of claim 8, wherein the
dividing component includes means for dividing using a plurality of
orthogonal basis functions.
15. A method for converting a received analog signal into a digital
signal, the method comprising: receiving an analog signal;
projecting the received analog signal over basis functions yielding
parallel data streams; sampling a plurality of basis coefficients;
calculating N basis coefficients in parallel analog computations
every T.sub.c seconds followed by N parallel analog to digital
converters each of which runs at a speed inversely proportional to
a time-window of T.sub.c seconds duration, introducing a trade-off
between sampling speed reduction and system complexity; associating
each of the N basis coefficients with a voltage; and quantizing the
voltage associated with at least one of the plurality of basis
coefficients at the end of the time window T.sub.c using a vector
quantizer with relaxed characteristics operating over a digital
conversion level that can change with a speed lower than that
required for time domain analog to digital conversion.
16. The method of claim 15 wherein potential lower bit requirements
can be achieved by optimally allocating a total finite number of
bits used in the quantification of the coefficients obtained
through the projection of the continuous-time signal over the basis
set in terms of number of bits per sample.
17. The method of claim 15 wherein a distortion introduced by the
analog to digital converter generated by a potential limited number
of coefficients N and by a total finite number of bits used in the
quantization of the coefficients reaches a zero value for a number
of coefficients N*.
18. The method of claim 15 wherein an optimal bit allocation per
coefficient among N coefficients is achieved when an output of the
analog to digital converter is a constant B.
19. The method of claim 15 wherein the number of basis coefficients
is reduced while keeping a level of distortion that is equal to a
level of distortion obtained in a time-domain analog to digital
conversion with pulse code modulation.
20. The method of claim 15 wherein distortion gain is traded for a
reduction in number of parallel analog to digital converters, so
that the number of basis coefficients required is calculated from a
selected acceptable gain and the desired number of total finite
bits.
Description
BACKGROUND
Digital wideband (WB) systems (including
ultra-wideband--UWB--systems) are highly desirable, offering
flexibility and programmability. However, while system data rates
and bandwidths continue to expand, analog-to-digital converters
(ADCs) are limited in bandwidth, resolution, and power consumption.
Currently available architectures used in the fabrication of ADCs
include flash architecture, which is based on parallel techniques
that use 2.sup.b-1 comparators to achieve "b" bits of resolution.
While comparators can sample analog input signals simultaneously,
this can increase the speed of the flash ADC. Because of the
parallelism of this architecture, the number of comparators can
grow exponentially with b, thus increasing the power consumption
and also the circuitry area. This can facilitate an increase in the
input capacitance limiting the system bandwidth, thereby increasing
the difficulty to match components.
Some variations of flash architecture such as the folded-flash,
pipelined and time interleaved architectures have been proposed in
order to overcome some of these problems. However these techniques
are also not without difficulties that have slowed the evolution of
ADCs such as aperture jitter or aperture uncertainty, which is the
sample-to-sample variation of the instant in time at which sampling
occurs. Moreover, the speed of sampling can be limited by the
frequency characteristic of the device used in the design, which
can limit the ability of the comparators to make an unambiguous
decision about the input voltage.
To overcome these problems, techniques that aim to relax the
operational conditions of the ADC have been proposed.
Low-resolution ADC is possible with sigma-delta modulation. The
noise penalty associated with the use of a few bits or less in the
quantization process is overcome in the sigma-delta scheme by using
either signal oversampling or multi-band processing techniques. In
particular, when a single bit is used, the implementation is
greatly simplified and practical mono-bit WB (or UWB) digital
communications receivers have significant potential. These
techniques generally utilize sampling at or above the Nyquist rate
over the full signal bandwidth, and can therefore suffer from the
aforementioned high-speed issues. In addition, they provide a
single WB (or UWB) serial data stream, which may stress the digital
signal processing following the ADC.
An alternative is to channelize the analog signal by means of a
bank of bandpass filters, and the output of each filter are sampled
in parallel. Multi-rate approaches can also be used. ADC thus can
occur at a reduced rate for each of the resultant bandpass signals.
The bandpass outputs can also be frequency translated to baseband,
allowing the use of a single lowpass filter design. However, the
bandpass analog filter bank design is difficult, and the resulting
non-ideal filters cause signal leakage across the bands that can
degrade overall system performance unless properly accounted for.
The design of analog filters with sharp roll-off needed in the
multi-band ADC approaches also suffers from power consumption and
large circuitry area to accommodate the passive elements (i.e.,
inductors and capacitors). Additionally, implementation of the bank
of bandpass filters in the multi-band processing ideas can be
potentially troublesome; problems such as spectrum sharing due to
the non-ideal characteristics of the bandpass filters can affect
the overall system performance.
High-speed signal processing utilized in analog to digital (A/D)
conversion of wideband signals (and ultra-wideband signals) imposes
challenging implementation problems, and sometimes impractical
power consumption. Current time-domain A/D conversion encounters
technological barriers as time-domain signal features shrink to
very fine resolution, on the order of tenths, and sometimes
hundredths of nanoseconds. In order to overcome these problems,
techniques that aim to relax the speed of the A/D conversion have
been proposed. In general, these techniques perform multi-band
signal processing in which the spectrum of the signal is
channelized into several bands by means of a bank of bandpass
filters. A/D conversion thus occurs at a much reduced speed for
each one of the resultant bandpass signals.
Further, a bank of frequency modulators can be used to shift the
signal spectrum so that the center frequency of each sub-band tends
toward zero frequency, allowing the use of a bank of identical
low-pass filters. Sigma-delta modulation has also been proposed
since it enables A/D conversion with low-resolution. The noise
penalty associated with the use of few bits in the quantization
process is overcome in the sigma-delta scheme by using either
signal oversampling or multi-band processing techniques. In
particular, when a single bit is used, the implementation is
greatly simplified and practical mono-bit digital receivers can be
implemented. Since all these techniques are based on time-domain
A/D conversion, they suffer from high-speed limitations, making it
desirable to channelize the signal spectrum into several
sub-bands.
Thus, a heretofore-unaddressed need exists in the industry to
address the aforementioned deficiencies and inadequacies.
SUMMARY
Included in this disclosure are methods for converting analog
signals to digital signals. At least one embodiment of the method,
among others includes receiving an analog signal; periodically
dividing the received analog signal into a plurality of discrete
signals at a predetermined interval, wherein each of the plurality
of divided signals is associated with a voltage; and quantizing the
voltage associated with at least one of the plurality of divided
signals.
Additionally included herein are embodiments of an analog to
digital converter. At least one embodiment of an analog to digital
converter includes at least one receiving component configured to
receiving an analog signal; at least one dividing component
configured to divide the received analog signal into a plurality of
discrete signals at a predetermined interval, wherein each of the
plurality of divided signals is associated with a voltage; and at
least one quantizer configured to quantize at least one voltage
associated with the at least one of the plurality of divided
signals.
Also included within this disclosure are embodiments of a computer
readable medium. At least one embodiment of a computer readable
medium includes logic configured to receive an analog signal; logic
configured to periodically divide the received analog signal into a
plurality of discrete signals at a predetermined interval, wherein
each of the plurality of divided signals is associated with a
voltage; and logic configured to quantize at least one voltage
associated with the at least one of the plurality of divided
signals.
Other systems, methods, features, and advantages of this disclosure
will be or become apparent to one with skill in the art upon
examination of the following drawings and detailed description. It
is intended that all such additional systems, methods, features,
and advantages be included within this description and be within
the scope of the present disclosure.
BRIEF DESCRIPTION
Many aspects of the disclosure can be better understood with
reference to the following drawings. The components in the drawings
are not necessarily to scale, emphasis instead being placed upon
clearly illustrating the principles of the present disclosure.
Moreover, in the drawings, like reference numerals designate
corresponding parts throughout the several views. While several
embodiments are described in connection with these drawings, there
is no intent to limit the disclosure to the embodiment or
embodiments disclosed herein. On the contrary, the intent is to
cover all alternatives, modifications, and equivalents.
FIG. 1 is an exemplary block diagram of an analog to digital
converter for expanding a received signal using a set of orthogonal
basis functions.
FIG. 2 is an exemplary block diagram of an analog to digital
converter that is configured to expand a received signal using a
set of basis functions, similar to the block diagram from FIG.
1.
FIG. 3A is an exemplary block diagram of a mixed signal receiver
with an analog to digital conversion via signal expansion, similar
to the block diagram from FIG. 1.
FIG. 3B is an exemplary block diagram of a transmitted reference
receiver that can be configured to operate in a continuous
time-domain, similar to the block diagram from FIG. 3A.
FIG. 3C is an exemplary block diagram of a transmitted reference
receiver that is configured to operated in a mixed signal
implementation, similar to the block diagram from FIG. 3A.
FIG. 4 is an exemplary block diagram of a frequency domain analog
to digital converter, similar to the block diagram from FIG. 1.
FIG. 5A is an exemplary raised cosine shaped power spectrum of a
Gaussian source for analog to digital conversion, such as in the
analog to digital converter from FIG. 4.
FIG. 5B is an exemplary graphical comparison between a Mean Square
Error (MSE) distortion of a time domain Pulse Code Modulation (PCM)
analog to digital converter, such as the analog to digital
converter from FIG. 3B.
FIG. 5C is an exemplary graphical comparison of gain versus the
number of coefficients for an average number of bits in an analog
to digital conversion, such as the analog to digital converter from
FIG. 4.
FIG. 6A is an exemplary raised cosine shaped power spectrum of a
Gaussian source for analog to digital conversion, similar to the
power spectrum from FIG. 5A.
FIG. 6B is exemplary graphical comparison between a Mean Square
Error (MSE) distortion of a time domain Pulse Code Modulation (PCM)
analog to digital converter, similar to the comparison in FIG.
5B.
FIG. 6C is an exemplary graphical illustration of orthogonal space
analog to digital gain against the number of coefficients for a
plurality of average bit rates, similar to the graphical
representation from FIG. 5C.
FIG. 7A is an exemplary signal to noise and distortion ratio for a
frequency domain analog to digital converter, such as the analog to
digital converter from FIG. 4.
FIG. 7B is an exemplary two-dimensional representation of the graph
from FIG. 7A.
FIG. 8A is an additional exemplary signal to noise and distortion
ratio for a frequency domain analog to digital converter, such as
the analog to digital converter from FIG. 4.
FIG. 8B is an exemplary two-dimensional representation of the graph
from FIG. 8A.
FIG. 9A is yet another exemplary signal to noise and distortion
ratio for a frequency domain analog to digital converter, such as
the analog to digital converter from FIG. 4.
FIG. 9B is an exemplary two-dimensional representation of the graph
from FIG. 9A.
FIG. 10 is an exemplary output signal to noise ratio of a
mixed-signal multi carrier receiver, such as the analog to digital
converter from FIG. 4.
FIG. 11 is an exemplary graphical depiction of bit error rate
curves for a transmitted reference receiver, such as shown in FIG.
4.
DETAILED DESCRIPTION
Included herein are techniques to perform analog to digital (A/D)
conversion, based on the quantization of coefficients obtained by
the projection of a continuous time signal over an orthogonal
space. The new A/D techniques proposed herein are motivated by the
sampling of an input signal in domains which may lead to lower
levels of signal distortion and significantly less demanding A/D
conversion characteristics. As a nonlimiting example, much of the
A/D conversion is studied in the frequency domain, where samples of
the signal spectrum are taken such that no time aliasing occurs in
the discrete time version of the signal. One can show that a
frequency domain analog to digital converter (ADC) overcomes some
of the difficulties encountered in conventional time-domain methods
for A/D conversion of signals with large bandwidths (such as
wideband) and very large bandwidths, such as ultra-wideband (UWB)
signals. The discrete frequency samples can then be passed through
a vector quantizer with relaxed characteristics, operating over DC
level that can change with a speed that is much lower than that
required for time domain A/D conversion. Fundamental points of
merit in A/D conversion and important system trade-offs are
discussed for the proposed frequency domain ADC. As a nonlimiting
example, at least one embodiment considers a multi-carrier UWB
communications scheme.
Also included herein are approaches to analog to digital conversion
that can overcome limitations encountered in the implementation of
time-domain A/D converters. These approaches can exploit signal
representation in domains other than the classical time domain,
which can reduce the speed of comparators that make quantization of
the sampled signal, and potentially improve the distortion versus
average bit rate of the A/D conversion. As a nonlimiting example,
at least one embodiment discussed herein considers a frequency
domain ADC in which samples of the signal spectrum are taken at a
rate that guarantees no aliasing in the discrete-time signal
domain. As discussed in more detail below, the consequences of
carrying out the A/D conversion in new domains are discussed, and
fundamental figures of merit in analog to digital conversion are
analyzed.
One approach is to project the received signal over basis
functions, and then sample the basis coefficients. Representing the
signal in a domain other than the classical time-domain sampling
approach yields parallel data streams, and potentially improves the
distortion versus average bit rate in the sampled output. The
time-domain signal may be reconstructed via a linear digital
computation, or signal processing can be carried out directly with
the basis coefficients.
"N" basis coefficients can be calculated in a parallel analog
computation every T.sub.c seconds, followed by N parallel ADCs.
Thus, the ADCs can run at a speed that is inversely proportional to
the time-window duration T.sub.c, which can be properly designed to
meet the speeds allowed by the technology used in the
implementation. The speed reduction comes at the cost of the
implementation of the local basis function generators, mixers and
integrators needed to project the continuous-time signal onto the
set of basis functions. This introduces a trade-off between
sampling speed reduction and system complexity that is
characterized in this disclosure. Although similar reductions in
the speed of the quantizers could be achieved in the time domain by
using a time-interleaved bank of quantizers, synchronization
problems, the very fine time resolution in high-speed applications,
and the fact that all the ADCs see the full bandwidth of the input
signal makes it difficult to design the sample/hold circuitry and
causes the overall design to require significant power. In
addition, the signal expansion approach avoids the sharp rolloff
filter bank needed in multi-band ADC architectures. Mixing with
basis functions, followed by integration over a time window
required to project the signal, generally synthesizes a filter bank
with overlapping spectrum and smooth transitions. The relaxed
implementation requirements are a motivation (among others) for the
ideas discussed herein.
Potential lower bit requirements and/or the potential improvement
in the distortion of the ADC of signal expansions, can be achieved
by optimally allocating the available number of bits in the
quantization of the coefficients obtained through the projection of
the continuous-time signal over the basis set. The possibility of
efficiently allocating the available resources in terms of number
of bits per sample is a feature that is not available in
conventional time-domain ADC. Optimal bit allocation is possible in
the proposed A/D conversion scheme because some signal
characteristics that are hidden in the time-domain, such as power
spectral distribution, can now be explored by projecting the
continuous-time signal.
As a nonlimiting example, one can consider A/D conversion in the
frequency domain, in which samples of the signal transform are
taken at a rate that guarantees no aliasing in the discrete-time
signal domain. The discrete frequency samples are then quantized by
a set of quantizers operating over DC levels that change with a
rate that is much lower than the Nyquist rate needed in the
sampling of the time domain signal. Other domains, such as those
provided by the Hadamard, Walsh, Walsh-Fourier and Haar wavelet
transforms, are also potential candidates.
Perhaps, the closest related publication is sampling of signal
projections, and some information theory work on over complete
expansions, where quantization of the coefficients of redundant
expansions is carried out. These publications study improved
sampling techniques based mainly on vector quantization,
oversampling, and signal reconstruction algorithms.
FIG. 1 is an exemplary block diagram of an analog to digital
converter for expanding a received signal using a set of orthogonal
basis functions. More specifically, the block diagram depicted in
FIG. 1 shows a basic orthogonal expansion principle of a proposed
A/D conversion. The received signal s(t) is decomposed every T,
seconds into N components, which are obtained through the
projection over a set of orthogonal bases 102a, 102b, and 102c
.PHI..function..times. ##EQU00001## The coefficients
.times. ##EQU00002## are found as:
.function..PHI..function..intg..times..function..times..PHI..function..ti-
mes..times.d ##EQU00003##
where the mean square error criterion is used to approximate the
received signal s(t) in a T.sub.c second interval as follows
.function..times..times..times..PHI..function. ##EQU00004##
At the end of the conversion time T.sub.c, the coefficient
.times. ##EQU00005## reach a constant value that is sent to a set
of quantizers
.function..times. ##EQU00006## 104a, 104b, and 104c, which return
the digital words
.times. ##EQU00007## These digital values represent the output of
this analog to digital converter for the input signal in a T.sub.c
second interval. When a limited number of coefficients are used in
the A/D signal conversion, some distortion can be introduced. This
distortion plus the distortion introduced by the quantization
process constitute the overall distortion of the proposed A/D
conversion, which is analyzed in more detail below.
The distortion introduced by the A/D converter in orthogonal spaces
is generated by the potential limited number of coefficients N, and
by the finite number of bits used in the quantization of the
coefficients
##EQU00008## The first distortion introduces an error
(e(n)=s(t)-s(t)) in the reconstruction formula (A2), where the
coefficients
##EQU00009## are calculated as in (A1) in order to minimize the MSE
distortion. The distortion obtained with N coefficients can be
expressed as
.PHI..times..sigma. ##EQU00010## where E.sub.s,T.sub.c is the
energy of the signal in the conversion interval T.sub.c
.sigma. ##EQU00011## is the variance of the coefficient a.sub.l,
and the distortion D.sub..PHI.,N is nonnegative by definition. When
the distortion reaches the zero value for a number of coefficients
N*, then s(t)=s(t), where the equality holds in the sense that the
approximation error has zero energy.
The distortion introduced by the finite number of bits used in the
quantization of the orthogonal domain coefficients will be denoted
as D.sub..PHI.,Q that is commonly quantified by the average mean
square error (MSE). One can define
.times. ##EQU00012## and
.times. ##EQU00013## as the random variables associated to the
coefficients
.times. ##EQU00014## and
.times. ##EQU00015## respectively. Thus the MSE of the quantization
error is
.PHI..times..times..times..times..PHI..times..times. ##EQU00016##
where
.PHI..times..epsilon..PHI..times..sigma..PHI..times..times.
##EQU00017## for a sufficiently high bit rate R.sub.l, where
.epsilon..PHI. ##EQU00018## is a constant that depends on the
probability density function (pdf) of a.sub.l, namely p(a).
Therefore the average distortion introduced by the quantization
process is
.PHI..times..times..epsilon..PHI..times..sigma..PHI..times..times.
##EQU00019##
At this point, one may desire to find the optimal bit allocation
among the N coefficients, i.e. it is desirable to find the set of
rates
.times. ##EQU00020## constrained
.times..times. ##EQU00021## such that the distortion in (A5) is
minimized. This classical optimization problem can be solved using
Lagrange multipliers, leading to the following result
.times..epsilon..PHI..times..sigma..PHI..times..times..epsilon..PHI..time-
s..sigma..PHI. ##EQU00022## The optimum solution assigns more bits
to the coefficients with larger variance such that the distortion
of all the coefficients is uniform and equal to
.PHI..PHI..times..times..epsilon..PHI..times..sigma..PHI..times..times.
##EQU00023##
This bit allocation resembles the concept of reverse water filling
found in rate distortion theory. One should note that if the
variance of one coefficient is sufficiently small, the result rate
from Equation (A6) could be negative, which in practice would mean
that the coefficient can be discarded. It is interesting to compare
the performance of the newly defined A/D conversion with the
conventional pulse coding modulation (PCM) technique in which each
time-domain sample is quantized with the same number of bits R. The
distortion incurred by PCM is
.epsilon..times..sigma..times..times. ##EQU00024## where the
sub-index t stands for time, .epsilon..sub.t depends on the pdf of
any sample of the time-domain signal, which is assumed stationary,
and
.sigma. ##EQU00025## is the sample's variance. Now, one can define
a very important figure of merit of the proposed A/D conversion in
orthogonal spaces, the orthogonal space A/D conversion gain
(G.sub.OSADC). This nonlimiting example compares the performance of
the proposed A/D method with the performance of a conventional A/D
with PCM, which is defined as
.PHI..PHI. ##EQU00026## which is the ratio between the distortion
of PCM and the distortion introduced by both the limited number of
coefficients and the quantization error when carrying out the A/D
conversion in orthogonal spaces. After substitution, the equation
becomes
.epsilon..times..sigma..times..times..PI..times..epsilon..PHI..times..sig-
ma..PHI..times..times..times..times..sigma..PHI. ##EQU00027##
It is interesting to analyze the special case in which the number
of coefficients reaches the defined value N*, which makes zero the
distortion pointed out in (A3). In that case, (A9) can be expressed
as:
.epsilon..times..times..epsilon..PHI..times..times..times..times..sigma..-
PHI..times..times..sigma..PHI. ##EQU00028##
where the last equality follows from the fact that the average
energy of the coefficients
.times. ##EQU00029## equals the time sample's variance since the
error e(n) has zero energy. Equation (A10) shows the potential gain
of the proposed method as the G.sub.OSADC is proportional to the
ratio of the arithmetic mean of the orthogonal coefficients
variances to the geometric mean of the same variances. Since the
arithmetic mean is larger than or equal to the geometric mean,
being equal only when all the variances are the same, and in
general
.epsilon..ltoreq..times..times..epsilon..PHI. ##EQU00030## where
G.sub.OSADC.gtoreq.1 or D.sub.PCM.gtoreq.D.sub..PHI.,Q under the
same average bit rate. Notice that a more uneven distribution of
the variances leads to a larger gain, which can be advantageous in
domains where the variance distribution is known or can be
predicted.
The nature of this ADC leads to carrying out digital signal
processing (DSP) applications in the same domain used in the A/D
conversion itself. A classical example of this duality is the
time-frequency pair, which is hereafter studied in the context of
A/D conversion.
FIG. 2 is an exemplary block diagram of an analog to digital
converter that is configured to expand a received signal using a
set of basis functions, similar to the block diagram from FIG. 1.
More specifically, FIG. 2 illustrates a basic signal expansion
principle of a proposed A/D conversion. A received signal s(t) is
decomposed every T.sub.c seconds into N components, which are
obtained through the projection over the basis
.PHI..function..times. ##EQU00031## 202a, 202b, and 202c. The
coefficients
.times. ##EQU00032## are found as
.function..PHI..function..intg..times..function..times..PHI..function..ti-
mes..times.d ##EQU00033## as illustrated with reference elements
204a, 204b, 204c, and 204d.
If the mean square error (MSE) criterion is used to reconstruct the
received signal s(t) in the interval
mT.sub.c.ltoreq.t.ltoreq.(m+1)T.sub.c through a linear combination
of the basis functions
.PHI..function..times. ##EQU00034## in general the coefficients
.times. ##EQU00035## can be linearly transformed. One can define
the N.times.N matrix .PSI., which contains the correlation
coefficients of the basis functions,
.psi..PHI..function..PHI..function..intg..times..PHI..function..times..PH-
I..function..times..times.d.times. ##EQU00036##
The coefficients a.sub.l.sup.(m) that provide an MSE approximation,
can be found by solving the linear equation
s.sup.(m)=.PSI.a.sup.(m) (A13)
where the vectors s.sup.(m) and a.sup.(m) are defined as
s.sup.(m)=[s.sub.0.sup.(m) . . . s.sub.N-1.sup.(m)].sup.T and
a.sup.(m)=[a.sub.0.sup.(m) . . . a.sub.N-1.sup.(m)].sup.T. Solving
Equation (A13) requires invertibility of the matrix .PSI.. If the
basis functions are orthonormal, then a.sup.(m)=s.sub.(m). An MSE
approximation can be given by
.function..times..times..PHI..function..times..times..times..ltoreq..ltor-
eq. ##EQU00037##
where the signal s.sup.(m)(t), 0.ltoreq.t.ltoreq.T.sub.c, is an MSE
approximation of the input signal s(t),
mT.sub.c.ltoreq.t.ltoreq.(m+1)T.sub.c. At the end of the conversion
time T.sub.c, the coefficients
.times. ##EQU00038## reach a constant value that is sent to a set
of quantizers
.times. ##EQU00039## 206a, 206b, 206c, and 206d, one for each
coefficient, which can return the digital words
.times. ##EQU00040## The l.sup.th quantizer Q.sub.l.sup.(m) has
2.sup.b.sup.t output levels, where
.times. ##EQU00041## is the number of bits used to obtain the
quantized set of coefficients
.times. ##EQU00042## These values represent the output of the
analog to digital converter for the input signal in a T.sub.c
second interval. One should note that, in this nonlimiting example,
the signal s(t) is being segmented by a rectangular window for
simplicity, however windows with preferable characteristics can
instead be used. The number of coefficients N used in the A/D
conversion is intimately related to the conversion time T.sub.c,
and can affect the degree of the approximation indicated in
Equation (A14), up to the point where the signal s(t) is
represented with zero energy with a sufficient number of
coefficients N*
.PHI..function..times. ##EQU00043## The existence of the number N*
that makes the mean square error zero assumes that the basis
functions
.PHI..function..times. ##EQU00044## span the input signal s(t). It
is also possible that N* tends to infinity, as for example happens
with signals with infinite spectral support (non-bandlimited
signals) when they are projected in the frequency domain. However,
for simplicity in the analysis, it is assumed that the input signal
s(t) is a smooth, well behaved signal (in some embodiments a
bandlimited signal) that can be represented with a finite number of
coefficients N*. The particular conditions that s(t) can satisfy
for the existence of N* will depend on the domain chosen for the
A/D conversion.
When only a limited number of coefficients (N.ltoreq.N*) is used in
the A/D signal conversion, some distortion is introduced. This
distortion, plus the distortion introduced in the quantization
process, can constitute major sources of distortion of the proposed
A/D conversion, and is analyzed below. One can also consider timing
and frequency offset distortion, discussed in more detail
below.
Without loss of generality one can consider the interval
0.ltoreq.t.ltoreq.T.sub.c, in which the coefficients
.times. ##EQU00045## at the output of the A/D converter provide a
representation of the analog input signal in the conversion time
Tc. The reconstructed signal is expressed as
.function..times..times..PHI..function..times..ltoreq..ltoreq.
##EQU00046##
Using the MSE criterion, the total distortion D can be expressed
as
.times..times..function..function..times..function..function..times..fun-
ction..function..times..times..times..times..PHI..function..times..times..-
PHI..function..times..times..times..times..PHI..function..times..times..PH-
I..function..times..times..PHI..function..times..times..times..times..PHI.-
.function..times..times..times..PHI..function..times..times..times..times.-
.function..times..PHI..function..times..PHI..function..times..times..times-
..times..function..times..PHI..function..times..PHI..function.
##EQU00047##
In order to eliminate the time dependence in the distortion D, one
can take the time average as follows:
.times..times..intg..times..times..function..function..times..times.d.ti-
mes..times..times..times..PHI..function..times..times..times..PHI..functio-
n..times..times..times..times..function..times..times..intg..times..PHI..f-
unction..times..PHI..function..times..times.d
.times..times..times..times..function..times..times..intg..times..PHI..fu-
nction..times..PHI..function..times..times.d .times.
##EQU00048##
where the bar ".sup.-" on top of any variable or operator indicates
time average, for example
.times..times..intg..times..times..times..times.d ##EQU00049## Both
terms in the third line of Equation (A17) are equal to zero due to
the fact that the functions
.PHI..function..times. ##EQU00050## are orthogonal in the interval
0.ltoreq.t.ltoreq.Tc. Thus, the total distortion introduced by the
A/D converter in orthogonal spaces is the sum of the truncation
distortion due to the potentially limited number of coefficients N,
which is denoted as D.sub.N, and the quantization distortion due to
the finite number of bits used in the quantization of the
coefficients
.times. ##EQU00051## which is denoted as D.sub.Q. One can restrict
the ADC distortion analysis presented in this analysis to signal
projection over orthonormal basis functions for simplicity of the
results. However, projection over linearly dependent basis
functions can also be employed.
The first distortion D.sub.N introduces truncation error in the
reconstruction formula (4), which can be expressed as
.function..function..function..function..times..times..PHI..function.
##EQU00052## where the coefficients
.times. ##EQU00053## are calculated as in Equation (A11) in order
to minimize the MSE distortion, i.e., to minimize the energy of the
error e(n). The distortion D.sub.N, obtained with N coefficients,
can be expressed as
.times..times..times..times..PHI..function..times..intg..times..times..ti-
mes..times..PHI..function..times..times.d.times..intg..times..times..times-
..times..times..times..PHI..function..times..PHI..function..times..times.d-
.times..times..times..times..times..times..intg..times..PHI..function..tim-
es..PHI..function..times..times.d.times..times..times..times..times..times-
..sigma..times..sigma..times..sigma..times..times..times..sigma.
##EQU00054##
where
.times..sigma. ##EQU00055## is the energy of the signal in the
conversion interval
.sigma. ##EQU00056## is the variance of the coefficient a.sub.l (it
is assumed E{a.sub.l}=0 for convenience) and the distortion D.sub.N
is nonnegative by definition. The fourth line in Equation (A19)
assumes that the functions
.PHI..function..times. ##EQU00057## are orthonormal. When the
number of coefficients is N*, the distortion reaches the zero value
and the received signal s(t) can be represented as
.function..times..times..PHI..function..ltoreq..ltoreq.
##EQU00058## where the quality holds in the sense that the
approximation error has zero energy. From a theoretical point of
view the truncation error e(n) can be made as small as desired,
however in a practical application this error may be non-zero as
the number of coefficients N can be limited by system constraints
such as complexity and circuitry area. In this case, the
coefficients with the largest variance
.sigma. ##EQU00059## can be chosen in order to minimize the error
energy in Equation (A19).
The distortion introduced by the finite number of bits used in the
orthogonal-domain quantization of the coefficients, D.sub.Q.cndot.,
is called quantization error and is commonly measured by the
average MSE, given
.times..times..times..times..PHI..function..times..times.
##EQU00060## where the same argument used in Equation (A19) is used
here to simplify the expression, and
D.sub.Q.sub.l=E{(a.sub.l-a.sub.l).sup.2}. A general closed-form
expression for the individual distortions D.sub.Q.sub.l has proven
difficult to find except for Gaussian sources, however for large
number of bits b.sub.l, an expression has been found as
.function..epsilon..times..sigma..times..times. ##EQU00061##
where
.epsilon. ##EQU00062## is a constant that depends on the
probability density function (pdf) of a.sub.l, namely p.sub.l(a),
as follows:
.epsilon..times..intg..infin..infin..times..function..times..times.d
##EQU00063## where {tilde over
(p)}.sub.l(a)=.sigma..sub.lp.sub.l(.sigma..sub.la). Therefore the
average distortion introduced by the quantization process can be
represented by
.times..times..times..epsilon..times..sigma..times..times.
##EQU00064## where the division by N is used to average across the
coefficients.
At this point, one can find an optimal bit allocation among the N
coefficients when the desired average number of bits per
coefficient at the output of the ADC is a constant B, (i.e., find
the set of number of bits
.times. ##EQU00065## constrained to
.times. ##EQU00066## such that the distortion in Equation (A24) is
minimized). This optimization problem can be solved using Lagrange
multipliers as shown below, leading to
.times..times..epsilon..times..sigma..times..times..epsilon..times..sigma-
. ##EQU00067##
Similar to above, this solution assigns more bits to the
coefficients with larger variance in order to make the distortion
of all the coefficients uniform and equal to
.times..times..times..times..epsilon..times..sigma..times..times..ltoreq.
##EQU00068##
This bit allocation is equivalent to the concept of reverse
water-filling found in rate distortion theory. One should note that
if the variance of one coefficient is sufficiently small, the
resultant number of bits from Equation (A25) could be negative,
which could mean that the coefficient could be discarded.
Additionally, the optimal solution in Equation (A25) can lead to a
fractional number of bits b.sub.l, which can be rounded off for
practical application. This optimal bit allocation can be used in
which the signal is linearly transformed to another domain before
reducing its resolution in order to make more efficient its
transmission over a communication channel. This technique is called
blockbased transform coding and assumes that either the original
digital data was obtained by means of high-resolution conventional
A/D conversion at the Nyquist rate or did not require A/D
conversion to be generated. This assumption constrains the
utilization of the block-based transform coding to discrete
applications where high-speed and high-resolution A/D conversion is
not required. On the other hand, at least one the techniques
discussed herein can be directly intended for A/D conversion since
it performs sampling in the same domain where the quantization
process is carried out.
One can also compare the performance of the A/D conversion based on
signal projection with the conventional pulse code modulation (PCM)
technique in which each time-domain sample is quantized with a
constant number of bits B. The average distortion incurred in PCM
assuming that N samples taken in a T.sub.c window comply with the
Nyquist criteria can be represented by
.times..epsilon..times..sigma..times..times. ##EQU00069## where the
sub-index t stands for time, .epsilon..sub.t depends on the pdf of
a sample of the time-domain signal which is assumed stationary,
.sigma. ##EQU00070## is the sample's variance, and B is large so
that this expression holds in general.
Now, one can define a figure of merit of the proposed A/D
conversion in orthogonal spaces, the orthogonal space A/D
conversion versus time domain PCM A/D conversion gain (G). This
nonlimiting example compares the performance of the proposed A/D
method with the performance of an A/D conversion with PCM, defined
as
##EQU00071##
Letting G* be the gain obtained with the optimum number of
coefficients N*,
##EQU00072## which is the ratio between the distortion of PCM and
the distortion introduced by both the limited number of
coefficients and the quantization error when carrying out the A/D
conversion via signal expansion. Substituting Equations (9), (11)
and (17) into Equation (A28), yields
.epsilon..times..sigma..times..times..times..times..times..epsilon..times-
..sigma..times..times..times..times..sigma..ltoreq..epsilon..times..sigma.-
.times..times..times..times..epsilon..times..sigma..times..times..times..t-
imes..sigma. ##EQU00073##
One can then analyze the case in which the number of coefficients
reaches the defined value N*, which makes D.sub.N zero. In this
case, Equation (20) can be expressed as:
.epsilon..times..sigma..times..times..times..times..times..epsilon..times-
..sigma..times..times..ltoreq..epsilon..times..sigma..times..times..times.-
.times..epsilon..times..sigma..times..times..epsilon..times..times..epsilo-
n..times..times..times..sigma..times..times..sigma. ##EQU00074##
where the last equality follows from the fact that the average
energy of the coefficients
.times. ##EQU00075## equals the time sample's variance since the
error in Equation (A18) has zero energy. Equation (A31) shows the
potential gain of a proposed method, as the orthogonal space A/D
conversion gain is proportional to the ratio between the arithmetic
mean of the orthogonal coefficients variances and the geometric
mean of the same variances.
Since the arithmetic mean is generally greater than or equal to the
geometric mean, being equal only when all the variances are the
same, and in general
.epsilon..gtoreq..times..times..epsilon. ##EQU00076## one can see
that G*.gtoreq.1 or D.sub.PCM.gtoreq.{hacek over (D)}.sub.Q*, under
the same average number of bits B. One should note that a more
uneven distribution of the variances leads to a larger gain, which
can be advantageous in domains where the variance distribution is
known or can be estimated.
One can assume that instead of being interested in taking advantage
of the reduction in distortion offered by the A/D conversion via
basis expansion, some embodiments can reduce the number of basis
coefficients while keeping a level of distortion that is equal to
the one obtained in time-domain A/D conversion with PCM, i.e., G=1.
This means that in some embodiments, trading distortion gain for
reduction in the number of parallel ADCs is desired. The question
that arises here is how many coefficients N.sub.G.sub.u are desired
to obtain a unit gain (G.sub.u=G=1) for a desired average number of
bits B. Since symbolically solving for N.sub.G.sub.u in terms of B
from Equation (20) can be difficult, one can instead express B in
terms of N.sub.G.sub.u as follows
.function..times..epsilon..times..sigma..times..times..epsilon..times..si-
gma..times..sigma. ##EQU00077##
This means that if the gain G obtained in Equation (A30) is greater
than 1 for some N.ltoreq.N*, one can reduce the number of
coefficients to N.sub.G.sub.u, leading to an implementation that
requires fewer ADCs than the number of time-interleaved ADCs needed
to achieve the same distortion and the same sampling rate.
Additionally, the synchronization challenges that appear in the
time-interleaved ADC architectures due to the very fine time
resolution, are relaxed in the orthogonal space ADC because the
signal is quantized at the end of the time projection window
T.sub.c, which will be larger than the Nyquist period. One should
note that the distortion gain or the reduction in the number of
quantizers come at the cost of the alternative implementation of
the circuitry needed in the projection of the input signal over the
orthogonal basis.
The results presented thus far are suboptimal in the sense that the
distortion measure is based on scalar quantization of individual
coefficients, and only the bit distribution has been optimized.
Better performance can be attained if the distortion measure is
optimized jointly over all the coefficients, a concept known as
vector quantization. However, practical ADCs can be implemented
using scalar quantization in order to keep low levels of
complexity. Optimal bit allocation together with scalar
quantization provides an interesting gain as shown in Equation
(A31), with a reasonable compromise in system complexity.
At this point one can ask whether there exists an orthogonal space
that provides the best A/D conversion measured in MSE distortion.
In other words, whether one can find a set of orthogonal
functions
.PHI..function..times. ##EQU00078## that expand an input signal
s(t) leading to a minimum MSE for a given average number of bits
B.
Intuitively, the set of orthogonal basis functions can provide a
compact representation, so the number of coefficients N desired to
achieve some level of distortion is minimal. Additionally, from
Equation (A31), the optimal orthogonal space can minimize the
geometric mean of the variances of coefficients. One could try to
find a set of optimal orthogonal functions by setting up a
constrained optimization problem. However, this nonlimiting example
instead provides a discussion based on analogy with known results
in linear transformations of discrete-time signals. One should note
that in linear filter theory, when the Karhunen-Loeve transform
(KLT) is applied to a zero mean, wide-sense stationary random input
vector, the resultant output is a vector of uncorrelated random
variables, i.e., the KLT diagonalizes the auto-correlation matrix
of the discrete random process. This observation implies that the
KLT provides a very compact representation of the input signal.
This result can be proven by showing that the geometric mean of the
variances of coefficients is minimized when the coefficients are
obtained through the KLT. Unfortunately, the KLT can be difficult
to use in practice as it generally desires signal stationarity, and
the eigenvectors that constitute the basis of the transformation
are available when the statistics of the input signal are known,
conditions that are not easily met in real systems. In general, it
is desirable for a practical application to have an orthogonal
space that is signal independent. To this end, the frequency domain
provides a well understood orthogonal space for A/D conversion
based on signal projection, which is described in more detail
below.
One should note that, while the frequency domain is discussed
herein, this is a nonlimiting example of a domain that can be used
to achieve a desired result. Other domains may also be used,
depending on the desired configuration.
The ADC via signal expansion introduces a time delay as the signal
information of the last T.sub.c seconds is transferred to a new
domain and condensed into N coefficients. This latency should be
properly chosen according to the specific application. For
instance, in a communications system, a proper choice of T.sub.c
could be a number less than or equal to the transmitted symbol
period T. Additionally, the nature of this ADC leads to carrying
out digital signal processing (DSP) applications in the same domain
used in the A/D conversion itself.
As a nonlimiting example of a practical application of the ideas
presented herein (and with reference to element 102a from FIG. 3A),
one can investigate the design of mixed-signal communications
receivers based on the ADC ideas presented in this disclosure.
Embodiments of receivers can include mixed-signals in the sense
that in their analog front end, signal projection over basis
functions is performed before the parallel ADCs are applied.
Additionally, the information bits can be detected through a
discrete matched filter operation that takes place in the domain on
which the received signal has been expanded.
More specifically, one can assume that the signal s(t) is
transmitted over a linear communication channel with impulse
response h(t) r(t)=s(t)*h(t)+z(t),0.ltoreq.t.ltoreq.T. (A33) where
"*" indicates continuous-time convolution and z(t) is additive
white Gaussian noise (AWGN). In a typical conventional all-digital
linear communication receivers, the received continuous-time signal
is first passed through a time-domain A/D converter running at
Nyquist rate, and the discrete-time samples are then demodulated by
performing a discrete-time linear filtering operation. The
following analysis presents a fundamentally different approach for
the implementation of all-digital linear receivers, based on the
coefficients obtained from A/D conversion after signal
expansion.
Assume that the transmitted signal s(t) conveys the information
symbol a. In order to obtain an estimate of the transmitted symbol
from the set of basis coefficients, one can begin by expressing the
receiver structure as a linear filtering problem in the time
domain
.function..function..times..intg..times..tau..function..tau..times..funct-
ion..tau..times.d.tau. ##EQU00079## where a is the symbol estimate,
p(t) is the impulse response of the linear filter demodulator which
can be a simple matched filter, a RAKE receiver, an MMSE receiver,
etc. The output of this filter is sampled at t=T. For convenience,
Equation (A34) can be expressed as
.intg..times..function..tau..times..function..tau..times.d.tau..intg..tim-
es..function..tau..times..function..tau..times.d.tau. ##EQU00080##
where one can define g.sup.+(.tau.)=p(T-.tau.). Now, one can
proceed to segment the symbol duration time T into M time-slots of
duration T.sub.c. One can also define the following signals
r.sub.m(t)=r(t)w.sub.m(t) (A36) g.sub.m(t)=g(t)w.sub.m(t),
(A37)
For M=0, . . . , M-1 and the window w.sub.m(t) introduced in
Equation (A37) can be selected as rectangular for simplicity of the
analysis, although as mentioned earlier other windows with desired
characteristics could be used instead.
Using these definitions, the linear receiver output in (A34) can be
expressed as
.times..intg..times..times..function..tau..times..function..tau..times.d-
.tau..times..intg..times..times..function..tau..times..function..tau..time-
s.d.tau..times..intg..infin..infin..times..function..tau..times..function.-
.tau..times.d.tau. ##EQU00081## in which the integral in Equation
(A34) has been segmented into M integrals that run over intervals
of duration T.sub.c each, such that T=MT.sub.c.
In order to express the matched filter operations in the new
conversion domain, the signal expansion over the basis functions
.PHI..sub.l(t) is used to represent both the segmented received
signal and segmented receive filter, leading to
.times..times..intg..infin..infin..times..function..tau..times..function-
..tau..times.d.tau..times..times..intg..infin..infin..times..infin..times.-
.function..times..PHI..function..tau..times..infin..times..function..times-
..PHI..function..tau..times.d.tau..times..times..infin..times..infin..time-
s..function..times..function..times..intg..infin..infin..times..PHI..funct-
ion..tau..times..PHI..function..tau..times.d.tau..times..times..infin..tim-
es..infin..times..function..times..function..times..psi..apprxeq..times..t-
imes..times..times..function..times..function..times..psi.
##EQU00082## where
.function..times. ##EQU00083## and
.function..times. ##EQU00084## are the best MSE coefficients
representation as explained in Equation (A14), which can include
reversing the linear transformation of Equation (A13). One should
note that the series expansion in Equation (A39) has been
truncated, leading to some degree of error. Although this
truncation error should in principle degrade the receiver
performance, as shown below, any desired performance can be
achieved if the trade off between complexity in terms of number of
coefficients N, and sampling speed .DELTA.F.sub.c=1/T.sub.c=M/T, is
adequately set up. One should also note that if the basis functions
are orthonormal, (A39) reduces to
.apprxeq..times..times..times..times..function..times..function.
##EQU00085## which reduced the complexity of detection. The
trade-off between the choice of the basis functions, complexity of
the detection formula, and the degree of truncation error is
fundamental in the receiver design. FIG. 3A illustrates the
mixed-signal receiver architecture.
FIG. 3A is an exemplary block diagram of a mixed signal receiver
with an analog to digital conversion via signal expansion, similar
to the block diagram from FIG. 1. More specifically, the frequency
domain emerges as an appealing domain for the analog to digital
conversion of signals with very large bandwidth, since in principle
it relaxes the extremely fine time resolution needed in time-domain
ADCs, and provides a scalable architecture. The timing requirements
are relaxed since the sampling is performed at a rate that is lower
than the one imposed by Nyquist criteria.
As illustrated in FIG. 3A, the receiver can include a first
combiner 302 and a low pass filter 306. Also included are a
plurality of second combiners 306a, 306b, and 306c and a plurality
of integration function 308a, 308b, 308c, and 308d, which can
facilitate division of a received analog signal. Additionally
included ire a plurality of analog to digital converters 310a,
310b, 310c, and 310d, and summing logic 312, which can provide a
desired signal a.
FIG. 3B is an exemplary block diagram of a transmitted reference
receiver that can be configured to operate in a continuous
time-domain, similar to the block diagram from FIG. 3A. In this
nonlimiting example, one can consider the design of a mixed-signal
multi-carrier receiver based on "UWB Mixed-Signal Transform-Domain
Receiver Front-End Architectures", written by Sebastian Hoyos and
Brian Sadler, which is hereby incorporated by reference in its
entirety. Additionally incorporated by reference in their
entireties are "High-speed A/D Conversion for Ultra-Wideband
Signals Based on Signal Projection Over Basis Functions" by Hoyos,
Sadler, and Arce; "Analog to Digital Conversion of Ultra-Wideband
Signals in Orthogonal Spaces" by Hoyos, Sadler, and Arce; and
"Ultra-Wideband Analog-to-Digital Conversion Via Signal Expansion"
by Hoyos and Sadler.
With specific reference to FIG. 3B, a continuous time domain
version of a transmitted-reference receiver can include a band pass
filter 320, an analog to digital converter 324, a multiplier 328,
and a summer 330.
FIG. 3C is an exemplary block diagram of a transmitted reference
receiver that is configured to operated in a mixed signal
implementation, similar to the block diagram from FIG. 3.A. More
specifically, included in the mixed signal implementation of a
transmitted-reference receiver is a first combiner 350, a low pass
filter 352, a plurality of second combiners 354a, 354b, and 354c.
Also included are integration functions 356a, 356b, 356c, and 356d,
analog to digital converters 358a, 358b, 358c, and 358d.
Exponential functions 360a, 360b, 360c, and 360d are also included,
as well as third combiners a summer 364 and a fourth combiner
366.
FIG. 4 is an exemplary block diagram of a frequency domain analog
to digital converter, similar to the block diagram from FIG. 1.
More specifically, FIG. 4 shows a block diagram of the frequency
domain ADC in which the complex exponential functions of t at 402a,
402b, and 402c (see also 404a, 404b, and 404c) that constitute the
orthogonal basis allow sampling of the continuous-time signal at
the frequencies
.times. ##EQU00086## leading to the set of frequency
coefficients
.intg..times..function..times.e.times..times..times..times.d.times.
##EQU00087##
These coefficients can then be quantized by a set of quantizers
408a, 408b, 408c, and 408d
.times. ##EQU00088## which in turn produce the ADC output digital
coefficients
.times. ##EQU00089## The frequency sample spacing
.DELTA.F=F.sub.l-F.sub.l-1 complies with
.DELTA..times..times..ltoreq. ##EQU00090## in order to avoid
aliasing in the discrete-time domain. Thus, the optimal number of
coefficients N* necessary to fully sample the signal spectrum with
bandwidth W, without introducing time aliasing, is proportional to
the time-bandwidth product
.DELTA..times..times..gtoreq. ##EQU00091## where the operator .left
brkt-top..right brkt-bot. is used to ensure that N* is the closest
upper integer that avoids discrete-time aliasing. Because signals
found in applications are time-limited, the term bandwidth here
refers to the range of frequencies in which the signal power is
larger than some defined power level, for instance, many signal
bandwidths are defined at 3 dB of attenuation, although more
conservative attenuation could be desirable for some applications
such as A/D conversion. Moreover, the bandwidth W described herein
is the bandwidth of the time-segmented signal which in general is
larger than the bandwidth of the signal s(t), as the segmentation
introduces side-lobes that can be sampled in order to obtain lower
distortion error. When
.DELTA..times..times. ##EQU00092## Equation (A42) becomes an
equality and the discrete-time alias-free condition is satisfied
without oversampling of the frequency spectrum.
One should note that the frequency-domain ADC can be fundamentally
different from the ADC architectures based on filter bank theory.
The difference lies in the fact that the frequency domain ADC
samples the expansion of time-segments of the received signal,
whereas the filter bank approach performs frequency channelization.
The computation of Fourier samples, via mixing and integration, can
be thought of as synthesizing a filter bank. However, these filters
are very broad with smooth transitions. In fact, the frequency
response of the branches in FIG. 3A can overlap with each other,
but this overlapping does not introduce ADC distortion. On the
other hand, the filter bank approach generally requires filters
with very sharp roll-off and any leakage between the channels will
seriously degrade the ADC performance. This is an important
motivation for the implementation of the frequency domain ADC.
FIG. 5A is an exemplary raised cosine shaped power spectrum of a
Gaussian source for analog to digital conversion, such as in the
analog to digital converter from FIG. 4. More specifically, as a
nonlimiting example, one can consider a stationary zero-mean
Gaussian continuous source with variance
.sigma. ##EQU00093## bandwidth W=2 GHz @ -55 dB, central frequency
F.sub.c=7 GHz and a power spectrum density (PSD) shown in FIG. 5A.
Notice that in this case the signal PSD provides the information
about the coefficient's variance distribution needed in the optimal
bit allocation. The signal is segmented in intervals of T.sub.c=3
ns to be A/D converted, thus requiring a frequency spacing between
samples of .DELTA.F=1/T.sub.c=333.33 MHz to avoid aliasing in the
discrete-time equivalent signal. The bits are optimally distributed
among the coefficients as indicated by Equation (15), leading to
the set of curves of MSE distortion versus average number of bits B
discussed below.
FIG. 5B is an exemplary graphical comparison between a Mean Square
Error (MSE) distortion of a time domain Pulse Code Modulation (PCM)
analog to digital converter, such as the analog to digital
converter from FIG. 3B. More specifically, MSE distortion for PCM
is shown for comparison purposes.
FIG. 5C is an exemplary graphical comparison of gain versus the
number of coefficients for an average number of bits in an analog
to digital conversion, such as the analog to digital converter from
FIG. 4. The A/D conversion gain (G) is plotted in FIG. 5C against
the number of coefficients N (N*=7) for several values of average
number of bits B. These drawings show the potential gain of
performing the A/D conversion in the frequency domain together with
optimal bit allocation, especially when the target average number
of bits is low. For this nonlimiting example, a gain of up to 3.35
(5.25 dB) can be achieved when N*=7 coefficients are
implemented.
Further, assume that a mono-bit (i.e., B=1) implementation is
desired and lowering the sampling rate is the main concern in the
design. So, one may desire to trade distortion gain for a lower
sampling rate of the ADCs. FIG. 5C shows that a mono-bit
implementation with N=5 coefficients (9 real-valued ADCs since the
DC-frequency coefficient requires just 1 ADC while the other
complex-valued frequency samples require 2 ADCs) can achieve the
same distortion rate of a time-domain ADC with PCM. However, the
frequency-domain ADC operates at 1=T.sub.c=333:33 MHz whereas the
time-domain ADC requires a sampling rate of 4 GHz to meet the
Nyquist criterion. If a time-interleaved architecture is
implemented in the time domain ADC to reduce the speed of the
comparators to 333.33 MHz, a total of 12 ADCs can be used, leading
to an implementation that requires 3 more ADCs than the frequency
domain ADC implementation. One should note that although the MSE
distortion in Equation (A22) is generally valid for a large number
of bits, for Gaussian sources the expression holds asymptotically
even for a small number of bits, so the curves in FIGS. 5A, 5B, and
5C are exact under the assumptions of this example.
Additionally, one can calculate the signal to noise and distortion
error for the frequency domain ADC that includes gain distortions,
timing errors, frequency offsets and additive noise in all the
paths of the ADC. It is assumed that a testing sinusoidal signal
r(t)=A cos(2.pi.F.sub.xt) is driven to the ADC input. The frequency
samples provided by the ADC under distortion and noise can be
expressed as
.function..times..intg..times..function..DELTA..times..times..times.e.pi.-
.function..DELTA..times..times..times..times..times.d.times.
##EQU00094##
where g.sub.n is the gain distortion, .DELTA.t.sub.n is the time
error, .DELTA.F.sub.n is the frequency offset and o.sub.n is the
additive noise associated with the nth frequency sample. For the
specific case of a test signal r(t)=A cos(2.pi.F.sub.xt) and using
the identity
cos(2.pi.F.sub.x(t-.DELTA.t.sub.n))=cos(2.pi.F.sub.xt)cos(2.pi.F.sub.x.DE-
LTA.t.sub.n)+sin(2.pi.F.sub.xt)sin(2.pi.F.sub.x.DELTA.t.sub.n),
Equation (A43) reduces to
.function..times..times..intg..times..function..times..pi..times..times..-
function..DELTA..times..times..times.e.pi..function..DELTA..times..times..-
times..times..times.d.times..times..function..times..pi..times..times..tim-
es..DELTA..times..times..times..intg..times..function..times..pi..times..t-
imes..times..times.e.pi..function..DELTA..times..times..times..times..time-
s.d.times..times..times..pi..times..times..times..DELTA..times..times..tim-
es..intg..times..function..times..pi..times..times..times..times.e.pi..fun-
ction..DELTA..times..times..times..times..times.d.times..function..times..-
pi..times..times..times..DELTA..times..times..times..intg..times..function-
..times..pi..times..times..times..times.e.pi..function..DELTA..times..time-
s..times..times..times.d.times..function..times..pi..times..times..times..-
DELTA..times..times..times..intg..times..function..times..pi..times..times-
..times..times.e.pi..function..DELTA..times..times..times..times..times.d.-
times..times..function..DELTA..times..times..times.e.pi..times..times..tim-
es..DELTA..times..times. ##EQU00095## Wherein R(F.sub.n) is given
by
.function..function..pi..times..times..function..pi..times..times..functi-
on. ##EQU00096## It can be noted from Equation (A45) that if there
are no frequency offsets, only the sample R(F.sub.x) will be
nonzero since all the other samples align with the nulls of
Equation (A45).
The SNDR can be given by the ratio of the power at the frequency of
interest F.sub.x, and the sum of the powers at all the other
frequencies F.sub.n. One should note that the phase rotation of the
sample {tilde over (R)}(F.sub.z) due to time jitter will degrade
the SNDR. To see this, one can express {tilde over (R)}(F.sub.x) in
real and imaginary parts {tilde over
(R)}(F.sub.x)=Ag.sub.xR(F.sub.n+.DELTA.F.sub.x)cos(2.pi.F.sub.x.DELTA.t.s-
ub.x)+o.sub.xR+j*(Ag.sub.xR(F.sub.n+.DELTA.F.sub.x)sin(2.pi.F.sub.x.DELTA.-
t.sub.x)+o.sub.xl) (A46) where o.sub.xR=Re{o.sub.x} and
o.sub.xI=Im{o.sub.x}. The real part of {tilde over (R)}(F.sub.x)
corresponds to the signal component whereas the imaginary part of
{tilde over (R)}(F.sub.x) introduces distortion, where the amount
of jitter .DELTA.t.sub.x determines how much distortion is
introduced. Thus, the SNDR is given by
.times..times..times..function..noteq..times..times..times..function..tim-
es..times..times..function. ##EQU00097## where the orthogonality
among the frequency samples allows an expression of an expected
value of the sum as a sum of expected values. The numerator in
Equation (A47) is the signal power component, which as shown below
can be approximated as
.times..times..function..apprxeq..times..sigma..function..pi..times..sigm-
a..DELTA..times..times..times..times..pi..times..sigma..DELTA..times..time-
s..times..sigma. ##EQU00098## where
.sigma..times..times..sigma..DELTA..times..times..times..DELTA..times..ti-
mes..DELTA..times..times..times..times..times..times..times..sigma..DELTA.-
.times..times..times..times..DELTA..times..times. ##EQU00099##
which assumes that the distortions in all the paths have the same
second order moments. Notice that the parameters
.sigma..DELTA..times..times. ##EQU00100## and
.sigma..DELTA..times..times. ##EQU00101## have included the
normalization factors
.DELTA..times..times. ##EQU00102## and
##EQU00103## respectively.
The noise terms in the sum of the denominator of Equation (A47) are
shown below to be
.times..function..apprxeq..times..sigma..DELTA..times..times..times..sigm-
a..function..sigma. ##EQU00104## The noise term due to the
imaginary part of {tilde over (R)}(F.sub.x) is also approximated
below as
.times..times..function..apprxeq..times..sigma..function..pi..times..sigm-
a..DELTA..times..times..times..times..pi..times..sigma..DELTA..times..time-
s..times..sigma. ##EQU00105## These results lead to the following
expression for the SNDR,
.times..times..sigma..function..pi..times..sigma..DELTA..times..times..ti-
mes..times..pi..times..sigma..DELTA..times..times..times..sigma..times..si-
gma..sigma..DELTA..times..times..times..noteq..times..times..pi..times..si-
gma..DELTA..times..times..times..times..pi..times..sigma..DELTA..times..ti-
mes..times..sigma. ##EQU00106## This expression reveals several
interesting results. For instance, the gain variance has little or
no effect on the SNDR if the additive noise second order moment is
zero, which in practice means little impact on the SNDR for
reasonable values of
.sigma. ##EQU00107##
Setting
.sigma..sigma..DELTA..times..times. ##EQU00108## and
.sigma. ##EQU00109## yields the SNDR due to frequency offset
.times..function..pi..times..sigma..DELTA..times..times..sigma..DELTA..ti-
mes..times..times..noteq..times..times. ##EQU00110## With
.sigma..sigma..DELTA..times..times. ##EQU00111## and
.sigma. ##EQU00112## one can find the SNDR due to time jitter
.times..function..times..pi..times..sigma..DELTA..times..times..times..pi-
..times..sigma..DELTA..times..times. ##EQU00113## and setting
.sigma..sigma..DELTA..times..times. ##EQU00114## and
.sigma..DELTA..times..times. ##EQU00115## one can find the SNDR due
to amplitude offset
.times..function. ##EQU00116##
FIG. 6A is an exemplary raised cosine shaped power spectrum of a
Gaussian source for analog to digital conversion, similar to the
power spectrum from FIG. 5A. As illustrated, a plot of
coefficient's variance distribution versus frequency is plotted.
The variance scale ranges from zero to 10, and the frequency (in
GHz) ranges from zero to 10.
FIG. 6B is exemplary graphical comparison between a Mean Square
Error (MSE) distortion of a time domain Pulse Code Modulation (PCM)
analog to digital converter, similar to the comparison in FIG. 5B.
More specifically, Mean Square Error distribution is plotted
against average bit rate for various values of N. As illustrated N
can take the value of 1, 3, 13, 15, and 17. Additionally included
in this nonlimiting example is a plot of PCM in time domain. One
should note that N can be any integer value. Although the
description with regard to FIG. 6B illustrates n as including
values of 1, 3, 13, 15, and 17, these are nonlimiting examples, as
N is a flexible design parameter that, depending on the desired
configuration can take any integer value.
FIG. 6C is an exemplary graphical illustration of a orthogonal
space analog to digital gain against the number of coefficients for
a plurality of average bit rates, similar to the graphical
representation from FIG. 5C. As illustrated, R can take the value
of 1, 3, 5, 8, and 10. Additionally, this nonlimiting example
illustrates that gain converges at 10.sup.2, when the number of
coefficients=17. One should note that R can be any integer value.
Although the description with regard to FIG. 6C illustrates n as
including values of 1, 3, 5, 8, and 10, these are nonlimiting
examples, as above.
With regard to FIGS. 6A-6C, the frequency domain emerges as an
appealing domain for the analog to digital conversion of signals
with very large bandwidths since it relaxes the extremely fine time
resolutions needed in time-domain ADCs. As described above, FIG. 4
shows the block diagram of the frequency domain ADC in which
samples of the continuous-time signal spectrum at the
frequencies
.times. ##EQU00117## are taken, leading to the set of frequency
coefficients
##EQU00118##
These coefficients are then quantized by a set of quantizers
.function..times. ##EQU00119## that, in turn, produce the ADC
output digital coefficients
.times. ##EQU00120## The frequency sample spacing
.DELTA.F=F.sub.l=F.sub.l-1 complies with
.DELTA..times..times..ltoreq. ##EQU00121## in order to avoid
aliasing in the discrete-time domain. Thus, the minimum number of
coefficients N necessary to sample a signal spectrum with bandwidth
W, without introducing time aliasing, is proportional to the
time-bandwidth product
.DELTA..times..times..gtoreq. ##EQU00122##
Assuming that the input signal s(t) is real-valued, the
coefficients
.times. ##EQU00123## are generally related to the-discrete Fourier
transform (DFT) coefficients as follows
.function..function..function..function..function. ##EQU00124##
where one can assume that samples are taken from 0 Hz; "*" denotes
the complex conjugate, F, =K/T.sub.c, is the time sampling
frequency and k=0, . . . , K-1. So, if N samples are taken from the
spectrum of the real-valued continuous-time signal, K=2(N-1) DFT
coefficients are obtained and these coefficients constitute the
digital representation of the time-domain signal. The relationship
in Equation (A57) is valid when the K samples of the continuous
time signal in the interval T.sub.c comply simultaneously with
Nyquist rate (F.sub.s.gtoreq.2W) and Equation (56). Otherwise, the
samples
.function..times. ##EQU00125## provide an approximate
representation of the signal.
As a nonlimiting example, one can consider a stationary zero-mean
Gaussian source with sample variance at and a power spectrum
density (PSD) that follows a raised cosine shape with rolloff
factor .alpha.=1, center frequency F.sub.c=7 GHz and bandwidth 4
GHz as shown in FIG. 6A. The signal is segmented in intervals of
T.sub.c=3 ns to be A/D converted, thus indicating a frequency space
between samples of AF=333.33 MHz. The bit rates can be distributed
among the coefficients, leading to the set of curves of MSE
distortion versus average bit rate R plotted in FIG. 6B. The MSE
distortion for PCM is also shown for comparison purposes. The
orthogonal space A/D conversion gain (G.sub.OSADC) is plotted in
FIG. 6C against the number of coefficients N for several values of
average bit rates R. These drawings show the potential gain of
performing the A/D conversion in the frequency domain together with
optimal bit allocation, especially when the target average bit rate
is low. For this example, a gain of up to 54.7 (5.2 dB) can be
achieved when N=17 coefficients can be implemented. Notice that
although the MSE distortion is in general valid for large bit
rates, for Gaussian sources the expression holds even for small
rates, so the curves in FIGS. 6A-6C are generally exact.
A/D conversion in orthogonal spaces can be utilized in numerous
applications including communications and signal processing
problems such as signal modulation, matched filtering and
space-time array processing. As a nonlimiting example, the matched
filtering problem can be carried out in the frequency domain thanks
to the time-frequency duality provided by Fourier analysis.
One can assume that the frequency domain ADC provides a set of full
resolution coefficients S.sub.m=[S.sub.m(0), . . . , S.sub.m(K-1)]
every T.sub.c, seconds, where m=0 . . . , M-1 and the information
symbol period T is related with the A/D conversion period as
T=MT.sub.c. One can begin with expressing the calculation of the
matched filter output {circumflex over (m)} in the time domain
.times..intg..times..function..tau..times..function..tau..times..times.d.-
tau. ##EQU00126## where h(t) is the impulse response of the matched
filter and the output of this filter is sampled at t=T. To reflect
the effect of segmenting the information symbol time T.sub.s into M
time-slots of duration T.sub.c, one can define the following
signals
.function..function..ltoreq..ltoreq..function..function..ltoreq..ltoreq.
##EQU00127## where m=0, . . . , M-1, and the signals s.sub.m(t) and
h.sub.m(t) are equal to zero outside the interval
0.ltoreq.t.ltoreq.T.sub.c. Using these definitions, the matched
filter output in Equation (A58) can be expressed as
.times..intg..times..times..times..tau..times..function..tau..times.d.tau-
..times..intg..times..function..tau..times..function..tau..times..times.d.-
tau..times..times. ##EQU00128## in which the integral in Equation
(A58) has been segmented into M pieces of duration T.sub.c each,
such that T=MT.sub.c.
In order to express the matched filter operations in the
frequency-domain, the Fourier transform is applied to Equation
(A60), which can result in
.times..times..times..intg..times..function..tau..times..function..tau..t-
imes..times.d.tau..times..times..intg..times..function..tau..times..functi-
on..tau..times..times.d.tau..times..intg..infin..infin..times..function..t-
imes..function..times..times.d ##EQU00129## where
S.sub.m(F)={s.sub.m(t)} and H.sub.m(F)={h.sub.m(T-t)} the second
line in Equation (A61) follows from the linearity property of the
Fourier transform and the third line follows from Parseval's
theorem. So, the exact calculation of the matched filter output in
the frequency domain can require the Fourier transforms of all the
segmented received signals, namely
.function..times. ##EQU00130## and the Fourier transform of the
segmented matched filters,
.function..times. ##EQU00131## However, since K samples of the
signal spectrum are provided by the frequency ADC, (A61) is can be
approximated as
.times..times..intg..infin..infin..times..function..times..function..time-
s..times.d.apprxeq..times..times..DELTA..times..times..times..times..funct-
ion..times..function..times..DELTA..times..times..times..times..times..fun-
ction..times..function. ##EQU00132## If the number of frequency
samples K is chosen such that the discrete-time signals
s.sub.m(n)=IDFT {S.sub.m(k)) and h.sub.m(n)=IDFT {H.sub.m(k)),
where n, k=0, . . . , K-1, comply with both no discrete-time
aliasing and Nyquist rate, the error introduced in Equation (18) is
negligible.
Expressing a signal as in Equation. (A1), where the
coefficients
##EQU00133## are information symbols to be transmitted through a
communications channel, opens a general technique to transmit
information using orthogonal signals. A nonlimiting example of
multi-orthogonal transmissions is the multi-carrier transmission
technique where the orthogonal functions are the complex
exponentials
.function..times..times.e.times..times..times..times..pi..times..times..t-
imes..times..ltoreq..ltoreq. ##EQU00134## As a nonlimiting example,
embodiments of this disclosure can include multi-carrier
modulation.
FIG. 7A is an exemplary signal to noise and distortion ratio for a
frequency domain analog to digital converter, such as the analog to
digital converter from FIG. 4. More specifically, FIG. 7A
illustrates the impact of the implementation impairments on the
SNDR. Each plot shows the impact of two impairments on the SNDR;
the left column shows the 3-dimensional plots and the right column
shows the corresponding isolines. The number of frequency samples
is N=5.
FIG. 7B is an exemplary two-dimensional representation of the graph
from FIG. 7A. More specifically, FIG. 7B illustrates normalized
time jitter versus normalized frequency offset.
FIG. 8A is an additional exemplary signal to noise and distortion
ratio for a frequency domain analog to digital converter, such as
the analog to digital converter from FIG. 4. More specifically,
FIG. 8A illustrates a graph of SNR versus additive noise and
normalized frequency offset.
FIG. 8B is an exemplary two-dimensional representation of the graph
from FIG. 8A. More specifically, FIG. 8B illustrates additive noise
versus normalized frequency offset.
FIG. 9A is yet another exemplary signal to noise and distortion
ratio for a frequency domain analog to digital converter, such as
the analog to digital converter from FIG. 4. More specifically FIG.
9A illustrates SNR versus normalized time jitter and additive
noise.
FIG. 9B is an exemplary two-dimensional representation of the graph
from FIG. 9A. More specifically, FIG. 9A illustrates normalized
time jitter versus additive noise.
To understand the implication of the results presented in FIGS. 7A
through 9B, one can consider a practical SNDR test scenario. One
can assume that a F.sub.x=5 GHz tone drives the frequency ADC. The
sampling period T.sub.c is chosen to be five periods of the
sinusoidal signal, T.sub.c=1 ns, which leads to a frequency spacing
between samples of .DELTA.F.sub.c=1 GHz. Additionally, the DC has
N=5 branches which implies that 5 samples of the spectrum are taken
for each T.sub.c sec window. Thus, the 5 samples are uniformly
distributed around the tone frequency leading to the frequency
range 3-7 (inclusive) GHz. If the frequency samples do not suffer
from frequency offset, only the sample at F.sub.x=5 can be nonzero
as the other samples will lie at the nulls of the spectrum of the
input signal. However, frequency offset can result in the
components collecting some undesired energy, degrading the SNDR as
defined in Equation (A47). To obtain a sense of practical levels of
frequency offset and timing errors that can be tolerated in this
specific example, assuming that from FIGS. 7A through 9B one can
conclude that acceptable SNDR value is around 40 dB. This is
achieved with a normalized frequency offset second moment
.sigma..DELTA..times..times. ##EQU00135## near 10.sup.-6 GHz.sup.2
together with a normalized time error with second moment around
10.sup.-6 s.sup.2. Then, the second order moment of the frequency
offset can be equal to
.times..DELTA..times..times..DELTA..times..times..times..sigma..DELTA..ti-
mes..times..times..times..times. ##EQU00136## Now, assuming a
deterministic frequency offset, a second order moment of 10.sup.12
GHz.sup.2 can be achieved by a fixed offset of 1 MHz or less. So,
for the highest oscillator frequency, i.e., the one at 10 GHz, a
requirement of only 103 parts per million (ppm) is desired.
In the case of timing errors, the target SNDR can be achieved with
normalized second moment of around 10.sup.-6, which leads to an
absolute time error second moment of
.times..DELTA..times..times..sigma..DELTA..times..times..times..times.
##EQU00137##
Once again assuming a fixed time offset, a second order moment of
4.times.10.sup.26 s is achieved with a time offset of 0.2
picoseconds (ps). Thus, up to 0.2 ps of time offset are allowed in
time windows of 1 nanosecond (ns) for the specific example
considered here. Lastly, an acceptable value of AWGN noise is
around 10.sup.-6 W/Hz.
FIG.
FIG. 10 is an exemplary output signal to noise ration of a
mixed-signal multi carrier receiver, such as the analog to digital
converter from FIG. 4. More specifically, FIG. 10 illustrates SNR
versus sampling rate for different values of SNR.sub.AWGN=10
log.sub.10(E.sub.b/N.sub.o).
This disclosure explores analog to digital conversion of signal
expansions, where an input analog signal is projected over a set of
basis functions before quantization takes place. Quantization can
be carried out over the coefficients obtained from this projection.
This A/D conversion technique provides a potential gain over
time-domain ADCs when optimal bit allocation is used in the
quantization process of the coefficients. Additionally, a reduction
of the sampling rate is achieved as the A/D conversion is performed
at the end of a properly chosen time window of T.sub.c seconds
during which the signal is projected. The sampling rate reduction
can be associated with an increase in the number of basis functions
on which the continuos-time signal is projected, leading to a
fundamental trade-off between complexity and sampling rate.
Further, this technique possesses some degree of flexibility in the
design as trading between speed and distortion can be achieved by
properly choosing the conversion time T.sub.c and the number of
coefficients N.
This disclosure also establishes a framework for a family of
mixed-signal communications receivers as a potential application of
the ADC framework discussed herein, and closed-form expressions for
symbol detection have been found. Embodiments of the receiver
disclosed in architecture provides a means to implement wide-band
communication receivers with parallel processing at lower sampling
speeds, without time-domain signal reconstruction.
As a nonlimiting example, the frequency domain can constitute an
appealing domain to perform the A/D conversion of signals that are
wideband, such as ultra-wideband signals. Moreover, having samples
of the signal spectrum can encourage implementing many
communications and signal processing applications in the frequency
domain. More specifically, one can show how the matched filter can
be implemented, even though segmentation of the time-domain signal
is used to reduce the number of coefficients. Additional robustness
is obtained by the ADC in the frequency domain as it naturally
filters narrow-band interference that lies away from the spectrum
points where samples are taken. As a nonlimiting example, in
communication systems using multi-carrier transmission, at least
one embodiment of the A/D conversion in the frequency domain goes
together with a very simple frequency domain implementation of the
digital correlators needed for the estimation of the information
symbols. Although the frequency domain is perhaps the oldest and
best understood domain besides the time-domain, other domains may
have desirable characteristics when carrying out A/D conversion. As
a nonlimiting example, from a circuit implementation point of view,
generation of the sinusoidal signals operating at frequencies
.times. ##EQU00138## used for the projection of the input signal in
the frequency domain ADC might lead to higher levels of complexity
and power consumption. Lower complexity can be achieved by
generating binary waveforms instead of sinusoidal ones.
Transformations that use orthogonal signals with binary waveforms
include the Hadamard transform, Walsh and Walsh-Fourier transform,
and the Haar wavelet transform.
Digital systems that interface with real-world signals, such as
voice, audio, communication waveforms, array processing etc. can be
implemented with the A/D conversion ideas discussed herein. The
solutions provided herein are especially beneficial for wideband
systems and ultra-wideband systems, such as communications and
geolocation.
Given a desired average number of bits B, one can allocate a total
number of bits NB among the N coefficients, so that the error in
Equation (A24) is minimized. The Lagrange multiplier method
provides a solution to this constrained optimization problem as
.function..times..times..epsilon..times..sigma..times..times..times..lamd-
a..function..times. ##EQU00139## where .lamda. must be chosen to
satisfy
.times. ##EQU00140## Now, setting to zero the derivative of
Equation (A64) with respect to b.sub.l leads to
.differential..differential..times..times..times..times..times..epsilon..-
times..sigma..times..times..times..lamda. ##EQU00141## .lamda. is
determined by the taking the product of Equation (A66) for all l,
leading to
.lamda..times..times..times..times..times..times..times..epsilon..times..-
sigma..times..times..times..times..times..times..times..times..times..time-
s..times..epsilon..times..sigma..times..times. ##EQU00142## so
that
.lamda..times..times..times..times..times..times..times..epsilon..times..-
sigma..times..times. ##EQU00143## After substitution in (A66), the
optimum bit allocation is found to be
.times..times..epsilon..times..sigma..times..times..epsilon..times..sigma-
. ##EQU00144##
For simplicity of the analysis one can assume that the impairments
.DELTA.F.sub.n, .DELTA.t.sub.n, and On have equal variance in all
the paths, i.e.,
.sigma..times..times..sigma..DELTA..times..times..times..DELTA..times..ti-
mes..DELTA..times..times..times..sigma..DELTA..times..times..times..times.-
.DELTA..times..times..times..times..times..sigma..times..times.
##EQU00145##
One should also note that the parameters
.sigma..DELTA..times..times. ##EQU00146## and
.sigma..DELTA..times..times. ##EQU00147## are normalized with the
factors
.DELTA..times..times. ##EQU00148## and
##EQU00149## respectively. The SNDR can be given by
.times..times..times..times..function..noteq..times..times..function..tim-
es..function. ##EQU00150##
The exact calculation of the expected values involved in Equation
(A70) can be difficult. A simpler and perhaps more meaningful
solution is provided by using truncated Taylor expansions. The
following Taylor expansions can be used
.function..apprxeq..times..times..times..times..times..times..function..a-
pprxeq..times..times..times..times..times..times..function..apprxeq..times-
..times..times..times..times..times. ##EQU00151##
The signal power term can be given by
.times..times..function..times..times..times..function..function..pi..tim-
es..times..function..DELTA..times..times..pi..times..times..function..DELT-
A..times..times..times..function..times..pi..times..times..times..DELTA..t-
imes..times..apprxeq..times..times..sigma..function..pi..times..sigma..DEL-
TA..times..times..times..times..pi..times..sigma..DELTA..times..times..tim-
es..sigma. ##EQU00152##
The noise terms in the sum of the denominator of Equation (A70) are
approximated as
.times..function..times..times..times..function..function..pi..times..tim-
es..function..DELTA..times..times..pi..times..times..function..DELTA..time-
s..times..times..apprxeq..times..times..sigma..DELTA..times..times..times.-
.sigma..function..sigma..noteq. ##EQU00153## The noise term due to
the imaginary part of {tilde over (R)}(F.sub.x) is approximated
as
.times..times..function..times..times..times..function..function..pi..tim-
es..times..function..DELTA..times..times..pi..times..times..function..DELT-
A..times..times..times..function..times..pi..times..times..times..DELTA..t-
imes..times..apprxeq..times..times..sigma..function..pi..times..sigma..DEL-
TA..times..times..times..times..pi..times..sigma..DELTA..times..times..tim-
es..sigma. ##EQU00154## These results lead to the following
expression for the SNDR
.times..times..sigma..function..pi..times..sigma..DELTA..times..times..ti-
mes..times..pi..times..sigma..DELTA..times..times..times..sigma..times..si-
gma..function..sigma..DELTA..times..times..times..noteq..times..pi..times.-
.sigma..DELTA..times..times..times..times..pi..times..sigma..DELTA..times.-
.times..times..sigma. ##EQU00155##
FIG. 11 is an exemplary graphical depiction of bit error rate (BER)
curves for a transmitted reference receiver, similar to that shown
in FIG. 4. More specifically, BER is plotted against SNR. A full
resolution curve, a mono-bit frequency domain curve, and a mono-bit
time domain curve is shown.
Generally, in terms of hardware architecture, embodiments of a
device that can perform functions described herein can include a
processor, volatile and nonvolatile memory, and one or more input
and/or output (I/O) device interface(s) that are communicatively
coupled via a local interface. The local interface can include, for
example but not limited to, one or more buses or other wired or
wireless connections. The local interface may have additional
elements, which are omitted for simplicity, such as controllers,
buffers (caches), drivers, repeaters, and receivers to enable
communications. Further, the local interface may include address,
control, and/or data connections to enable appropriate
communications among the aforementioned components. The processor
may be a hardware device for executing software, particularly
software stored in volatile and nonvolatile memory.
The processor can be any custom made or commercially available
processor, a central processing unit (CPU), an auxiliary processor
among several processors associated with the device, a
semiconductor based microprocessor (in the form of a microchip or
chip set), a macroprocessor, or generally any device for executing
software instructions. Examples of suitable commercially available
microprocessors are as follows: a PA-RISC series microprocessor
from Hewlett-Packard.RTM. Company, an 80.times.86 or Pentium.RTM.
series microprocessor from Intel.RTM. Corporation, a PowerPC.RTM.
microprocessor from IBM.RTM., a Sparc.RTM. microprocessor from Sun
Microsystems.RTM., Inc, or a 68xxx series microprocessor from
Motorola.RTM. Corporation.
The volatile and nonvolatile memory can include any one or
combination of volatile memory elements (e.g., random access memory
(RAM, such as DRAM, SRAM, SDRAM, etc.)) and nonvolatile memory
elements (e.g., ROM, hard drive, tape, CDROM, etc.). Moreover, the
memory may incorporate electronic, magnetic, optical, and/or other
types of storage media. Note that the volatile and nonvolatile
memory can have a distributed architecture, where various
components are situated remote from one another, but can be
accessed by the processor. Additionally volatile and nonvolatile
memory can include communications software and an operating
system.
The software in volatile and nonvolatile memory may include one or
more separate programs, each of which includes an ordered listing
of executable instructions for implementing logical functions.
Additionally, the software in the volatile and nonvolatile memory
may include an operating system. A nonexhaustive list of examples
of suitable commercially available operating systems is as follows:
(a) a Windows.RTM. operating system available from Microsoft.RTM.
Corporation; (b) a Netware.RTM. operating system available from
Novell.RTM., Inc.; (c) a Macintosh.RTM. operating system available
from Apple.RTM. Computer, Inc.; (d) a UNIX operating system, which
is available for purchase from many vendors, such as the
Hewlett-Packard.RTM. Company, Sun Microsystems.RTM., Inc., and
AT&T.RTM. Corporation; (e) a LINUX operating system, which is
freeware that is readily available on the Internet; (f) a run time
Vxworks.RTM. operating system from WindRiver.RTM. Systems, Inc.; or
(g) an appliance-based operating system, such as that implemented
in handheld computers or personal data assistants (PDAs) (e.g.,
PalmOS.RTM. available from Palm.RTM. Computing, Inc., and Windows
CE.RTM. available from Microsoft.RTM. Corporation). The operating
system essentially controls the execution of other computer
programs and provides scheduling, input-output control, file and
data management, memory management, and communication control and
related services.
A system component embodied as software may also be construed as a
source program, executable program (object code), script, or any
other entity comprising a set of instructions to be performed. When
constructed as a source program, the program is translated via a
compiler, assembler, interpreter, or the like, which may or may not
be included within the volatile and nonvolatile memory, so as to
operate properly in connection with the Operating System.
The Input/Output devices that may be coupled to system I/O
Interface(s) may include input devices, for example but not limited
to, a keyboard, mouse, scanner, microphone, etc. Further, the
Input/Output devices may also include output devices, for example
but not limited to, a printer, display, speaker, etc. Finally, the
Input/Output devices may further include devices that communicate
both as inputs and outputs, for instance but not limited to, a
modulator/demodulator (modem; for accessing another device, system,
or network), a radio frequency (RF) or other transceiver, a
telephonic interface, a bridge, a router, etc.
If the device is a personal computer, workstation, or the like, the
software in the volatile and nonvolatile memory may further include
a basic input output system (BIOS) (omitted for simplicity). The
BIOS is a set of software routines that initialize and test
hardware at startup, start the Operating System, and support the
transfer of data among the hardware devices. The BIOS is stored in
ROM so that the BIOS can be executed when the device is
activated.
When the device is in operation, the processor can be configured to
execute software stored within the volatile and nonvolatile memory,
to communicate data to and from the volatile and nonvolatile memory
and to generally control operations of the device pursuant to the
software. Software in memory, in whole or in part, are read by the
processor, perhaps buffered within the processor, and then
executed.
One should note that the flowcharts included herein show the
architecture, functionality, and operation of a possible
implementation of software. In this regard, each block can be
interpreted to represent a module, segment, or portion of code,
which comprises one or more executable instructions for
implementing the specified logical function(s). It should also be
noted that in some alternative implementations, the functions noted
in the blocks may occur out of the order. For example, two blocks
shown in succession may in fact be executed substantially
concurrently or the blocks may sometimes be executed in the reverse
order, depending upon the functionality involved.
One should note that any of the programs listed herein, which can
include an ordered listing of executable instructions for
implementing logical functions, can be embodied in any
computer-readable medium for use by or in connection with an
instruction execution system, apparatus, or device, such as a
computer-based system, processor-containing system, or other system
that can fetch the instructions from the instruction execution
system, apparatus, or device and execute the instructions. In the
context of this document, a "computer-readable medium" can be any
means that can contain, store, communicate, propagate, or transport
the program for use by or in connection with the instruction
execution system, apparatus, or device. The computer readable
medium can be, for example but not limited to, an electronic,
magnetic, optical, electromagnetic, infrared, or semiconductor
system, apparatus, or device. More specific examples (a
nonexhaustive list) of the computer-readable medium could include
an electrical connection (electronic) having one or more wires, a
portable computer diskette (magnetic), a random access memory (RAM)
(electronic), a read-only memory (ROM) (electronic), an erasable
programmable read-only memory (EPROM or Flash memory) (electronic),
an optical fiber (optical), and a portable compact disc read-only
memory (CDROM) (optical). In addition, the scope of the certain
embodiments of this disclosure can include embodying the
functionality described in logic embodied in hardware or
software-configured mediums.
It should be emphasized that the above-described embodiments are
merely possible examples of implementations, merely set forth for a
clear understanding of the principles of this disclosure. Many
variations and modifications may be made to the above-described
embodiment(s) without departing substantially from the spirit and
principles of the disclosure. All such modifications and variations
are intended to be included herein within the scope of this
disclosure.
* * * * *